The subject matter of the present invention relates to a simulator software method and apparatus, embodied in an earth formation reservoir simulator, for simulating an earth formation reservoir containing liquids and/or gases by solving a system of linear equations that characterize physical aspects of an oil and/or gas field, the amount of time required by the reservoir simulator to solve the system of linear equations and to thereby simulate the earth formation reservoir being reduced relative to prior art simulation methods practiced by prior art simulators.
Oil and gas is produced from underground rock formations. These rocks are porous, just like a sponge, and they are filled with fluid, usually water. This porous characteristic of rocks is known as porosity. These rocks in addition to being porous have the ability to allow fluids to flow through the pores, a characteristic measured by a property called permeability. When oil (or gas) is trapped in such formations, it may be possible to extract it by drilling wells that tap into the formation. As long as the pressure in the well is lower than that in the rock formation, the fluids contained in the pores will flow into the well. These fluids may then flow naturally up the well to the surface, or the flow up the well may have to be assisted by pumps. The relative amounts of oil, gas and water that are produced at the surface will depend on the fraction of the rock pore space that is occupied by each type of fluid. Water is always present in the pores, but it will not flow unless its volume fraction exceeds a threshold value that varies from one type of rock to another. Similarly, oil and gas will only flow as long as their volume fractions exceed their own thresholds.
The characteristics of the rock (including porosity and permeability) in an oil reservoir vary greatly from one location to another. As a result, the relative amounts of oil, gas and water that can be produced will also vary from reservoir to reservoir. These variations make it difficult to simply predict the amount of fluids and gases a reservoir will produce and the amount of resources it will require to produce from a particular reservoir. However, the parties interested in producing from a reservoir need to project the production of the reservoir with some accuracy in order to determine the feasibility of producing from that reservoir. Therefore, in order to accurately forecast production rates from all of the wells in a reservoir, it is necessary to build a detailed mathematical model of the reservoir's geology and geometry.
A large amount of research has been focused on the development of reservoir simulation tools. These tools include mathematical and computer models that describe and which are used to predict, the multiphase flow of oil and gas within a three dimensional underground formation (a "field"). Reservoir tools use empirically acquired data to describe a field. These data are combined with and manipulated by mathematical models whose output describes specified characteristics of the field at a future time and in terms of measurable quantities such as the production or injection rates of individual wells and groups of wells, the bottom hole or tubing head pressure at each well, and the distribution of pressure and fluid phases within the reservoir.
The mathematical model of a reservoir is typically done by dividing the reservoir volume into a large number of interconnected cells and estimating the average permeability, porosity and other rock properties for each cell. This process makes use of seismic data, well logs, and rock cores recovered when wells are drilled. Production from the reservoir can then be mathematically modeled by numerically solving a system of three or more nonlinear, partial differential equations describing fluid flow in the reservoir.
Computer analysis of production from an oil reservoir is usually divided into two phases, history matching and prediction. In the history matching phase, the past production behavior of the reservoir and its wells is repeatedly modeled, beginning with initial production and continuing up to the present time. The first computer run is based on a geological model as described above. After each run, the computer results are compared in detail with data gathered in the oil field during the entire period of production. Geoscientists modify the geological model of the reservoir on the basis of the differences between computed and actual production performance and rerun the computer model. This process continues until the mathematical reservoir model behaves like the real oil reservoir.
Once a suitable history match has been obtained, production from the oil reservoir can be predicted far into the future (sometimes for as long as 50 years). Oil recovery can be maximized and production costs minimized by comparing many alternative operating plans, each requiring a new run of the computer model. After a field development plan is put into action, the reservoir model may be periodically rerun and further tuned to improve its ability to match newly gathered production data.
When sufficient data is obtained about the reservoir, characteristics of a reservoir can be mathematically modeled to predict production rates from wells in that reservoir. The gross characteristics of the field include the porosity and permeability of the reservoir rocks, the thickness of the geological zones, the location and characteristics of geological faults, relative permeability and capillary pressure functions and such characteristics of the reservoir fluids as density, viscosity and phase equilibrum relationships. From this data, a set of continuous partial differential equations (PDEs) are generated that describe the behavior of the field as a function of time and production parameters. These production parameters include the locations of wells, the characteristics of the well's completions, and the operating constraints applied to the wells. Operating constraints may include such as the production rate of a particular fluid phase, the bottom hole pressure, the tubing head pressure, or the combined flow rates of a group of wells. These constraints may be applied directly by data or by means of another simulator that models the flow of fluids in the surface equipment used to transport the fluids produced from or injected into the wells. However, because only the simplest system of PDEs can be solved using classic or closed-form techniques (e.g., a homogeneous field having circular boundaries), a model's PDEs are converted into a set of non-linear approximations which are then solved numerically. One approximation technique is the finite difference method. In the finite difference method, reservoir PDEs are converted into a series of difference quotients which divide a reservoir into a collection of discrete three dimensional cells, which are then solved for at discrete times to determine (or predict) the value of reservoir characteristics such as pressure, permeability, fluid fractions, and at a later time.
Within the computerized reservoir simulator, reservoir performance is modeled in discrete increments of time. Each so-called timestep advances the solution from a previous point in time, where all variables are known, to a future point in time, where all variables are unknown. This process is repeated until the entire time period of interest has been modeled. Within each timestep it is necessary to solve a huge system of nonlinear equations that models fluid flow from cell to cell and through the wells. (With current technology it is possible to include several million cells in the reservoir model.) Solutions to the system of nonlinear equations are obtained by Newton iteration. In each such iteration the system of nonlinear equations is approximated by a system of linear equations, which must be solved by yet another iterative procedure.
A general outline of the operation of a reservoir simulator follows (refer to FIG. 8a). Reservoir data and rock core data are used to describe a computational grid and the properties of the reservoir rocks. This is combined with data on the physical properties of the fluids contained in the reservoir and used to compute the initial distributions of pressure and fluid saturations (volume fractions) as well as the composition of each fluid phase. Time varying data, such as the locations and characteristics of wells, production and injection flow rate controls, and simulator control information, is read from a data base. Using the current pressure, saturation, and fluid compositions for each grid cell, the partial differential equations describing mass balances are approximated by finite differences resulting in two or more nonlinear algebraic equations for each cell. In addition, these nonlinear equations are linearized by means of Newton's method. The resulting system of linear equations is solved iteratively using methods described in this specification. After the linear equations have been solved, a test is performed to determine whether all of the nonlinear terms in the finite difference equations have converged. If not, the simulator returns to a previous step wherein the partial differential equations describing mass balances are approximated by finite differences. However, if the nonlinear terms have converged, the simulator updates values to complete the current timestep. Then, the simulator tests to determine whether the desired ending time in the simulation has been reached. If not, the simulator returns to a previous step to read new time varying data and begins the next timestep. However, if the endpoint of the simulation has been reached, then, the simulator completes output operations and the run is finished.
As reservoir simulations grow in complexity (e.g., number of parameters) and size (e.g., number of cells), solving the resulting system of linear equations, represented by the matrix equation Ax=b requires an increasingly large effort. For example, the work involving the iterative solution increases with the square of the number of parameters per cell. In many reservoir models, the time required to solve systems of linear equations (step 56 in FIG. 8a) can be a limiting factor on the simulation's usefulness.
In connection with reservoir modeling, well logging operations are performed in the formation thereby producing well log data, and seismic operations are performed on the formation thereby producing seismic data. The seismic data is reduced thereby producing reduced seismic data. The well log data and the reduced seismic data are introduced, as input data, to a computer workstation which stores a gridding software and a simulator software. A gridding software, hereinafter known as "the Flogrid software" or the "Flogrid gridding software", is disclosed in prior pending U.S. patent application Ser. No. 09/034,701, filed in the U.S. on Mar. 4, 1998, which is based on a Great Britain patent application number 9727288.4 filed Dec. 24, 1997, the disclosure of which is incorporated by reference into this specification. The "Flogrid" gridding software includes another gridding software known as "Petragrid". The "Petragrid" gridding software is disclosed in prior pending U.S. patent application Ser. No. 08/873,234 filed Jun. 11, 1997, the disclosure of which is also incorporated by reference into this specification. The gridding software will respond to the reduced seismic data and the well log data by gridding the earth formation which was subjected to the well log operation and the seismic operation. The type of grids imposed on the earth formation include structured (approximately rectangular) grids and unstructured (tetrahedral) grids. A property, such as permeability or water saturation, is assigned to each cell or grid block of the grid. As a result, a set of output data is generated by the gridding software, the set of output data including the plurality of cells/grid blocks of the grid and the respective plurality of properties associated with each of the cells of the grid.
The set of output data from the gridding software are introduced, as input data, to a reservoir simulator software. The reservoir simulator software will respond to the set of output data from the gridding software by generating a plurality of simulation results which are associated, respectively, with the plurality of cells/grid blocks of the grid received from the gridding software. The plurality of simulation results and the plurality of cells/grid blocks associated therewith, generated by the reservoir simulator software, will be displayed on a 3D viewer of the workstation for observation by a workstation operator. Alternatively, the plurality of simulation results and the plurality of cells/grid blocks associated therewith can be recorded for observation by a workstation recorder.
The reservoir simulator software can model an oilfield reservoir. For example, in the Society of Petroleum Engineers (SPE) publication number 28545, concerning a transient tool for multiphase pipeline and well simulation, dated 1994, the authors have solved for pressure losses along a single pipeline using a technique related to conservation of material and conservation of pressure. A similar technique has been applied to a network of pipelines or flowlines in the Society of Petroleum Engineers (SPE) publication number 29125, authored by Litvak and Darlow. In this publication, the authors (Litvak and Darlow) have taken a network model (i.e., a network of pipelines) in which the pressure losses along the network branches can either be calculated from tables or from an analytical model, and the analytical model solves for three (3) conservations and pressures. In addition, in an article by the "Society of Petroleum Engineers" (SPE) 12259, each well being modeled in that article was characterized by three (3) variables: pressure, water fraction, and gas fraction.
As noted earlier, in many reservoir models, the time required to solve systems of linear equations (step 56 in FIG. 8a) can be a limiting factor on the simulation's usefulness. Accordingly, it would be beneficial to have a method that can be used to efficiently solve large systems of linear equations. It is the solution of this system of linear equations that is the subject of the invention of this specification.
Accordingly, a new and improved reservoir simulator software is needed wherein the amount of time required to execute the new and improved reservoir simulator software is much less than the amount of time required to execute the reservoir simulator software of the prior art.